FIFTH PROBLEM SHEET FOR ALGEBRAIC NUMBER THEORY
M3P15
AMBRUS PÁL
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√
1. In this problem ω = e2πi/3 = −1+2 −3 is a primitive 3-rd root of unity. Let
(x, y) be an integer solution of the Diophantine equation x2 + x + 1 = y 5 .
(a) Show that y is not divisible by 3.
(b) Prove that x − ω and x + 1 + ω are relatively prime in the ring E of Eisenstein
integers.
(c) Conclude that x − ω = (c + dω)5 in E, where c, d ∈ Z. Deduce that the only
solutions of this equation are x = √
0, y = 1 and x = −1, y = 1.
2. Compute the class group of Q( −10).
3. Let (x, y) be an integer solution of the Diophantine equation x2 + 10 = y 3 .
(a) Show that y must√be odd, and not divisible
by 5.
√
(b) Prove that (x + −10)
and
(x
−
−10)
are
relatively prime in the ring of
√
integers OK of K = Q(√ −10).
√
√
(c) Conclude that x + −10 is a cube, that is x + −10 = (c + d −10)3 in OK .
Deduce that the equation has no solutions.
Finally here is a problem to help you preparing for the mastery question in the
exam.
4. (a) Let p be an odd prime and let K be the splitting field of x2 + 1 over Fp .
What is the cardinality of K, depending on p?
(b) Let i ∈ K be a root of x2 + 1. Prove that
−1
2 p−1
1+i
= 1 + ip = (1 + i)p =
i 2 (1 + i)
p
p
in K, where ( ·· ) is the Legendre symbol.
(c) Deduce that if p is congruent to 1 mod 8, then 2 is a quadratic residue mod p.
Date: March 13, 2017.
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